Compute
\[\frac{1}{\cos 80^\circ} - \frac{\sqrt{3}}{\sin 80^\circ}.\]
Solution: First, we can write
\[\frac{1}{\cos 80^\circ} - \frac{\sqrt{3}}{\sin 80^\circ} = \frac{\sin 80^\circ - \sqrt{3} \cos 80^\circ}{\cos 80^\circ \sin 80^\circ}.\]From the angle subtraction formula, we can write the numerator as
\begin{align*}
\sin 80^\circ - \sqrt{3} \cos 80^\circ &= 2 \left( \frac{1}{2} \sin 80^\circ - \frac{\sqrt{3}}{2} \cos 80^\circ \right) \\
&= 2 (\cos 60^\circ \sin 80^\circ - \sin 60^\circ \cos 80^\circ) \\
&= 2 \sin (80^\circ - 60^\circ) \\
&= 2 \sin 20^\circ.
\end{align*}Also, from the angle addition formula, $\sin 160^\circ = \sin (80^\circ + 80^\circ) = \sin 80^\circ \cos 80^\circ + \cos 80^\circ \sin 80^\circ = 2 \cos 80^\circ \sin 80^\circ,$ so
\[\cos 80^\circ \sin 80^\circ = \frac{1}{2} \sin 160^\circ = \frac{1}{2} \sin 20^\circ.\]Therefore,
\[\frac{\sin 80^\circ - \sqrt{3} \cos 80^\circ}{\cos 80^\circ \sin 80^\circ} = \frac{2 \sin 20^\circ}{\frac{1}{2} \sin 20^\circ} = \boxed{4}.\]